R Exercise 4. Clustering and classification
I have loaded the Boston dataset that contains information on housing values in suburbs of Boston. Data has 506 observations and 14 differeent variables like crime rate of the town, number of rooms in dwelling, pupil-teacher ratio, proportion of the lower status population, etc. All these variables are the key that help to eveluate the value of the houses in this area.
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...## [1] 506 14The graphical representation of the variables show that in some cases there exist a strong correlation between variables; also the accumaulation tends to be close to the edge.
## function (x, ...)
## UseMethod("pairs")
## <bytecode: 0x7fc757bf6518>
## <environment: namespace:graphics>## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00However, I will also plot the correlation matrix in order to explore the data in more details.
## Warning: package 'tidyverse' was built under R version 3.3.2## Loading tidyverse: ggplot2
## Loading tidyverse: tibble
## Loading tidyverse: tidyr
## Loading tidyverse: readr
## Loading tidyverse: purrr
## Loading tidyverse: dplyr## Warning: package 'ggplot2' was built under R version 3.3.2## Warning: package 'tidyr' was built under R version 3.3.2## Conflicts with tidy packages ----------------------------------------------## filter(): dplyr, stats
## lag(): dplyr, stats
## select(): dplyr, MASS## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00
The correlogram is giving a more comprehensive picture of the correlation between the variables. Therefore, one can clearly observe a negative correlation between indus/dis (proportion of non-retail business acres per town to weighted mean of distances to five Boston employment centres), nox/dis (nitrogen oxides concentration to weighted mean of distances to five Boston employment centres), age/dis (proportion of owner-occupied units built prior to 1940 to weighted mean of distances to five Boston employment centres) and lstat/medv (lower status of the population to median value of owner-occupied homes). Positive correlation is observed in indus/nox (proportion of non-retail business acres per town to nitrogen oxides concentration), rad/tax (index of accessibility to radial highways to full-value property-tax rate per $10,000), nox/age. Having a closer look at the correlations, these seem to be logical.
However, in order to be able to classify the variable, it has to be standirdized so that it is comparable.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865## [1] "matrix"## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610Standardization of the variable has led to the fact that the range of the variable have decrease. Therefore, this standardized variable will be used in the further analysis
Also, I will create yet another categorical variable crime that will be created from the continuous one crim. I will remove crim variable from the dataset so that it does not affect the further analysis
## crime
## low med_low med_high high
## 127 126 126 127After the necessary transformation, I will divide the data by the train (contain 80% of the data) and the test one (20% of the data) in order to proceed with the Linear Discrimination Analysis.
Now, I will fit the linear discriminant analysis on the train set.I will use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. LDA will find a combination of the explanatory variable in such way so that it can separate the classes of the crime variable the best
## [1] "matrix"## crime
## low med_low med_high high
## 127 126 126 127## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2524752 0.2475248 0.2524752 0.2475248
##
## Group means:
## zn indus chas nox rm
## low 1.05906909 -0.8904696 -0.195131024 -0.8983270 0.4248582
## med_low -0.08376633 -0.2488762 0.042638951 -0.5857561 -0.1091553
## med_high -0.37626407 0.1687568 0.190859195 0.3701591 0.1672778
## high -0.48724019 1.0149946 0.003267949 1.0279369 -0.3725562
## age dis rad tax ptratio
## low -0.9407757 0.9245847 -0.6879988 -0.7181283 -0.44272466
## med_low -0.3745765 0.3412565 -0.5511961 -0.4646879 -0.03950843
## med_high 0.4854653 -0.3537503 -0.4639351 -0.3527581 -0.21131912
## high 0.8255786 -0.8450012 1.6596029 1.5294129 0.80577843
## black lstat medv
## low 0.3789132 -0.75985909 0.502133500
## med_low 0.3117072 -0.15366948 0.003391685
## med_high 0.1038478 0.04037345 0.170827168
## high -0.7220979 0.87102411 -0.644094735
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.08257361 0.63785558 -0.953093815
## indus 0.06544082 -0.15227820 0.322891573
## chas -0.09270221 -0.05534743 0.156489615
## nox 0.20379978 -0.82698298 -1.324012825
## rm -0.20607407 -0.19708916 -0.154700829
## age 0.19818808 -0.47509452 -0.241948571
## dis -0.09623188 -0.33783864 0.002967614
## rad 4.10376979 0.85820016 -0.072029213
## tax 0.02806966 0.13606484 0.516155991
## ptratio 0.13820802 -0.06341854 -0.301033032
## black -0.12630186 0.04911348 0.117523420
## lstat 0.12744620 -0.30200432 0.242435870
## medv 0.19639703 -0.40340161 -0.214018461
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9594 0.0316 0.0090
In order to see the full picture of the obtained results, I have plotted a graph where different colours are resposible for different clsses of the variables. The arrow indicates the impact of each of the predictor variable in the model. From that, one can clearly see that rad (index of accessibility to radial highways) has the longest arrow and respectively impact.
Now, I will remove the crime from the data and will make a prediction for the new dataset.
## crime
## low med_low med_high high
## 127 126 126 127## predicted
## correct low med_low med_high high
## low 12 13 4 0
## med_low 2 13 6 0
## med_high 0 9 17 3
## high 0 0 0 23The data show that prediction for the high and low crime rates are correct ones. However, the prediction for the medium crime rates are not always correct
Analysing the data from another angel, I will cluster observation and perform the k-means model that will asign cluster based on the distance between variables. Distance between the variablesis is a measure of its similarity.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.119 85.620 170.500 226.300 371.900 626.000
- When performing k-means model, I have set the number of the clusters equal to 10. Howevr, from the plot (Within Cluster Sum of Squares), it obvious that 2 clusters will enough due to the fact that after two clusters, the value drops significantly. So, I will adjust the model.*
The plot is showing the scaled pairs that are plotted against each other
- Also, I have decided to make a 3D plot I have fit the scaled train data that you used to fit the LDA.*
## [1] 404 13## [1] 13 3## Warning: package 'plotly' was built under R version 3.3.2##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutAlso, here is another 3D plot where the color is defined by the clusters of the k-mean. It shows the same classes as in LDA model, however, without classification by the crime rate.
As a bonus I have decide to perform k-means on the original Boston data. I have taken the clusters variable as the target class.
## crime
## low med_low med_high high
## 127 126 126 127## Call:
## lda(km2$cluster ~ ., data = Boston)
##
## Prior probabilities of groups:
## 1 2 3
## 0.1996047 0.5237154 0.2766798
##
## Group means:
## crim zn indus chas nox rm age
## 1 0.7491682 10.49505 12.800396 0.05940594 0.5798416 6.189772 73.15743
## 2 0.2323824 17.69811 6.666981 0.07547170 0.4831913 6.468596 55.55623
## 3 12.0799686 0.00000 18.397286 0.06428571 0.6719000 6.004857 89.91143
## dis rad tax ptratio black lstat medv
## 1 3.394095 4.801980 403.5743 17.73465 369.2717 12.875941 22.20693
## 2 4.867279 4.316981 276.0377 17.84943 388.9088 9.440679 25.97019
## 3 2.054707 22.878571 661.8357 20.12286 286.5699 18.572857 16.26143
##
## Coefficients of linear discriminants:
## LD1 LD2
## crim 0.001231960 -0.0061488634
## zn 0.009459824 -0.0014772000
## indus 0.028954393 -0.0204754827
## chas 0.083882416 -0.1920615150
## nox 1.688279455 3.5675807263
## rm -0.052904794 -0.0740868296
## age -0.003403119 -0.0009436915
## dis -0.133340151 -0.0756860539
## rad 0.128615007 -0.3498011613
## tax 0.021536890 0.0164145869
## ptratio 0.055921351 -0.0639644060
## black -0.002807248 0.0001461840
## lstat -0.001948397 -0.0002527576
## medv 0.009225193 0.0149027194
##
## Proportion of trace:
## LD1 LD2
## 0.971 0.029## Warning in arrows(x0 = 0, y0 = 0, x1 = myscale * heads[, choices[1]], y1 =
## myscale * : zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(x0 = 0, y0 = 0, x1 = myscale * heads[, choices[1]], y1 =
## myscale * : zero-length arrow is of indeterminate angle and so skipped
Variable nox (nitrogen oxides concentration) seems to be the most influential.